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Theorem ofcc 33720
Description: Left operation by a constant on a mixed operation with a constant. (Contributed by Thierry Arnoux, 31-Jan-2017.)
Hypotheses
Ref Expression
ofcc.1 (𝜑𝐴𝑉)
ofcc.2 (𝜑𝐵𝑊)
ofcc.3 (𝜑𝐶𝑋)
Assertion
Ref Expression
ofcc (𝜑 → ((𝐴 × {𝐵}) ∘f/c 𝑅𝐶) = (𝐴 × {(𝐵𝑅𝐶)}))

Proof of Theorem ofcc
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ofcc.2 . . . 4 (𝜑𝐵𝑊)
2 fnconstg 6780 . . . 4 (𝐵𝑊 → (𝐴 × {𝐵}) Fn 𝐴)
31, 2syl 17 . . 3 (𝜑 → (𝐴 × {𝐵}) Fn 𝐴)
4 ofcc.1 . . 3 (𝜑𝐴𝑉)
5 ofcc.3 . . 3 (𝜑𝐶𝑋)
6 fvconst2g 7209 . . . 4 ((𝐵𝑊𝑥𝐴) → ((𝐴 × {𝐵})‘𝑥) = 𝐵)
71, 6sylan 579 . . 3 ((𝜑𝑥𝐴) → ((𝐴 × {𝐵})‘𝑥) = 𝐵)
83, 4, 5, 7ofcfval 33712 . 2 (𝜑 → ((𝐴 × {𝐵}) ∘f/c 𝑅𝐶) = (𝑥𝐴 ↦ (𝐵𝑅𝐶)))
9 fconstmpt 5735 . 2 (𝐴 × {(𝐵𝑅𝐶)}) = (𝑥𝐴 ↦ (𝐵𝑅𝐶))
108, 9eqtr4di 2786 1 (𝜑 → ((𝐴 × {𝐵}) ∘f/c 𝑅𝐶) = (𝐴 × {(𝐵𝑅𝐶)}))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wcel 2099  {csn 4625  cmpt 5226   × cxp 5671   Fn wfn 6538  cfv 6543  (class class class)co 7415  f/c cofc 33709
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pr 5424
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-iun 4994  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7418  df-oprab 7419  df-mpo 7420  df-ofc 33710
This theorem is referenced by: (None)
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